The label \(\sigma = 1,2,3,4\)  takes four different values corresponding to flavors combining valleys and spin \((K_{+},\uparrow ), (K_{+},\downarrow ),(K_{-},\uparrow ),(K_{-},\downarrow )\) .
We neglect interactions between different valleys because they are described by the Coulomb interaction \(V ({\bf q} ; z)\)  with large wave vectors \({\bf q} \approx {\bf K}_{+} - {\bf K}_{-}\)  and this approximation treats the four flavors on an equal footing. Within a given flavor, the exchange potential breaks chiral symmetry and charge density is transferred between layers. In addition, the four flavors have a certain relative configuration.
By beginning the iterative procedure with different initial exchange profiles, we find different self-consistent solutions, and we evaluate the total energy of each solution in order to determine the ground state.
For example, for the AF state, exchange for a given flavor has odd parity with respect to spatial inversion, and the four flavors are arranged in an antiferromagnetic configuration so that there is no net charge polarization and, hence, no cost in terms of Hartree energy. Thus this state has lower energy at charge neutrality than, say, a ferrimagnetic or ferromagnetic configuration of the four flavors.
Since our approximation treats the four flavors equally, it is unable to differentiate three distinct combinations [1]}, [2]}, [3]} of spins and valleys within the antiferromagnetic configuration: layer-antiferromagnetic in which the polarization of flavors \((K_{+},\uparrow ),(K_{-},\uparrow )\)  is opposite to that of \((K_{+},\downarrow ),(K_{-},\downarrow )\) , quantum anomalous Hall when \((K_{+},\uparrow ), (K_{+},\downarrow )\)  are opposite to \((K_{-},\uparrow ),(K_{-},\downarrow )\) , or quantum spin Hall when \((K_{+},\uparrow )(K_{-},\downarrow )\)  are opposite to \((K_{+},\downarrow ),(K_{-},\uparrow )\) .
